Problem: $ C = \left[\begin{array}{rr}5 & 0 \\ 0 & 4\end{array}\right]$ $ D = \left[\begin{array}{rrr}5 & 4 & 5 \\ 2 & 2 & 4\end{array}\right]$ What is $ C D$ ?
Answer: Because $ C$ has dimensions $(2\times2)$ and $ D$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ C D = \left[\begin{array}{rr}{5} & {0} \\ {0} & {4}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{4} & \color{#9D38BD}{5} \\ {2} & \color{#DF0030}{2} & \color{#9D38BD}{4}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{5}\cdot{5}+{0}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{5}\cdot{5}+{0}\cdot{2} & ? & ? \\ {0}\cdot{5}+{4}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rrr}{5}\cdot{5}+{0}\cdot{2} & {5}\cdot\color{#DF0030}{4}+{0}\cdot\color{#DF0030}{2} & ? \\ {0}\cdot{5}+{4}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{5}\cdot{5}+{0}\cdot{2} & {5}\cdot\color{#DF0030}{4}+{0}\cdot\color{#DF0030}{2} & {5}\cdot\color{#9D38BD}{5}+{0}\cdot\color{#9D38BD}{4} \\ {0}\cdot{5}+{4}\cdot{2} & {0}\cdot\color{#DF0030}{4}+{4}\cdot\color{#DF0030}{2} & {0}\cdot\color{#9D38BD}{5}+{4}\cdot\color{#9D38BD}{4}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}25 & 20 & 25 \\ 8 & 8 & 16\end{array}\right] $